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Sep 20, 2013
09/13

by
Mirna Džamonja

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This is an article in mathematics, specifically in set theory. On the example of the Measure Recognition Problem (MRP) the article highlights the phenomenon of the utility of a multidisciplinary mathematical approach to a single mathematical problem, in particular the value of a set-theoretic analysis. MRP asks if for a given Boolean algebra $\algB$ and a property $\Phi$ of measures one can recognize by purely combinatorial means if $\algB$ supports a strictly positive measure with property...

Source: http://arxiv.org/abs/math/0608336v1

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Sep 20, 2013
09/13

by
Mirna Džamonja

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We prove that for regular $\lambda$ above a strong limit singular $\mu$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $\lambda$ and $\mu$ there are coboundedly many regular $\kappa

Source: http://arxiv.org/abs/math/0608638v1

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Sep 20, 2013
09/13

by
Mirna Džamonja

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We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using...

Source: http://arxiv.org/abs/math/0608636v1

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Jun 30, 2018
06/18

by
Mirna Džamonja

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We study certain Banach spaces that are added in the extension by one Cohen real. Specifically, we show that adding just one Cohen real to any model adds a Banach space of density $\aleph_1$ which does not embed into any such space in the ground model such a Banach space can be chosen to be UG This has consequences on the the isomorphic universality number for Banach spaces of density $\aleph_1$, which is hence equal to $\aleph_2$ in the standard Cohen model and the same is true for UG spaces....

Topics: Mathematics, Logic

Source: http://arxiv.org/abs/1409.8172

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Jun 30, 2018
06/18

by
Mirna Džamonja

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The article uses two examples to explore the statement that, contrary to the common wisdom, the properties of singular cardinals are actually more intuitive than those of the regular ones.

Topics: Mathematics, Logic

Source: http://arxiv.org/abs/1409.4940

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Sep 20, 2013
09/13

by
Mirna Džamonja

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This note contains a Stone-style representation theorem for compact Hausdorff spaces.

Source: http://arxiv.org/abs/math/0608384v1

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Sep 20, 2013
09/13

by
Mirna Džamonja

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We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element.

Source: http://arxiv.org/abs/math/0608330v1

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Sep 22, 2013
09/13

by
Mirna Dzamonja

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We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \[ \bigcup\{U(x): x\in D\}=X. \] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using...

Source: http://arxiv.org/abs/0811.1165v1

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Sep 19, 2013
09/13

by
MIrna Dzamonja; Istvan Juhasz

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We give a construction under $CH$ of a non-metrizable compact Hausdorff space $K$ such that any uncountable semi-biorthogonal sequence in $C(K)$ must be of a very specific kind. The space $K$ has many nice properties, such as being hereditarily separable, hereditarily Lindel\"of and a 2-to-1 continuous preimage of a metric space, and all Radon measures on $K$ are separable. However $K$ is not a Rosenthal compactum. We introduce the notion of bidiscrete systems in compact spaces and note...

Source: http://arxiv.org/abs/0910.3091v2

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Jul 20, 2013
07/13

by
Mirna Džamonja; Saharon Shelah

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This paper investigates a connection between the ordering triangleleft^ast among theories in model theory and the (N)SOP_n hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP_2 and SOP_1, and gives a strong connection between SOP_1 and the maximality in Keisler ordering. Together with the known results about the connection between the (N)SOP_n hierarchy and the existence of universal models in the absence of GCH, the paper provides a step...

Source: http://arxiv.org/abs/math/0009087v2

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Sep 24, 2013
09/13

by
Mirna Džamonja; Kenneth Kunen

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We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel...

Source: http://arxiv.org/abs/math/9408201v1

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Sep 18, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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Suppose that lambda = mu^+. We consider two aspects of the square property on subsets of lambda. First, we have results which show e.g. that for aleph_0

Source: http://arxiv.org/abs/math/9510216v1

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Jul 20, 2013
07/13

by
Mirna Džamonja; Saharon Shelah

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In this paper we investigate some properties of first order theories which prevent them from having universal models under certain cardinal arithmetic assumptions. Our results give a new syntactical condition, oak property, which is a sufficient condition for a theory not to have universal models in cardinality lambda when certain cardinal arithmetic assumptions implying the failure of GCH (and close to the failure of SCH) hold.

Source: http://arxiv.org/abs/math/0009078v1

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Sep 22, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as the name suggests, a weak version of stationary reflection. This sort of reflection was introduced in [DjSh:545] (math.LO/9601219), where it was shown that weak reflection has applications to various guessing principles, in the sense that if there is no weak reflection, than a guessing principle holds, and an application dealing with the saturation...

Source: http://arxiv.org/abs/math/0003118v1

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Sep 18, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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Various versions of club are shown to be different. A question of Soukup, Fuchino and Juhasz, is it consistent to have a stick without club, is answered as a consequence. The more detailed version of the paper, which is coming up, also answers a question of Galvin.

Source: http://arxiv.org/abs/math/9710215v1

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Sep 19, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists)...

Source: http://arxiv.org/abs/math/9911228v1

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Sep 18, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection

Source: http://arxiv.org/abs/math/9601219v1

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Sep 22, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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Suppose that $\lambda=\lambda^{ \lambda^{++}$. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind possible applications in analysis, we further observe that for such $\lambda$, for any fixed $\mu>\lambda^+$ regular with $\mu=\mu^{\lambda^+}$, it is consistent that $2^\lambda=\mu$ and there is no normed vector space...

Source: http://arxiv.org/abs/math/9805149v2

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Sep 19, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular...

Source: http://arxiv.org/abs/math/0102043v1

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Sep 22, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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We prove that club does not imply the existence of a Suslin tree, so answering a question of I. Juhasz.

Source: http://arxiv.org/abs/math/9612226v1

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Sep 19, 2013
09/13

by
Mirna Džamonja; Saharon Shelah

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This note gives two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular, while the second is a negative consistency result which applies to the situation of lambda singular. Both results are connected to an earlier result of the same authors in which they showed that a certain version of clubsuit holds at a successor of singular just in ZFC. The first result here shows that that fact can to a certain extent be extended...

Source: http://arxiv.org/abs/math/9709205v1

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Sep 20, 2013
09/13

by
Mirna Džamonja; Jean Larson

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We comment on a question of Justin Moore on colourings of pairs of nodes in an Aronszajn tree and solve an instance of it.

Source: http://arxiv.org/abs/math/0608382v1

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Sep 22, 2013
09/13

by
Joel David Hamkins; Mirna Džamonja

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If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

Source: http://arxiv.org/abs/math/0409304v1

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Sep 23, 2013
09/13

by
Piotr Borodulin-Nadzieja; Mirna Džamonja

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The paper investigates possible generalisations of Maharam's theorem to a classification of Boolean algebras that support a finitely additive measure. We prove that Boolean algebras that support a finitely additive non-atomic uniformly regular measure are metrically isomorphic to a subalgebras of the Jordan algebra with the Lebesgue measure. We give some partial analogues to be used for a classification of algebras that support a finitely additive non-atomic measure with a higher uniform...

Source: http://arxiv.org/abs/1105.1250v1

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Jun 30, 2018
06/18

by
James Cummings; Mirna Džamonja; Charles Morgan

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We prove that it is consistent that $\aleph_\omega$ is strong limit, $2^{\aleph_\omega}$ is large and the universality number for graphs on $\aleph_{\omega+1}$ is small. The proof uses Prikry forcing with interleaved collapsing.

Topics: Mathematics, Logic

Source: http://arxiv.org/abs/1408.4188

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Sep 18, 2013
09/13

by
James Cummings; Mirna Džamonja; Saharon Shelah

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In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say theta strongly non-reflects at lambda iff there is a function F: theta ---> lambda such that for all alpha < theta with cf(alpha)= lambda there is C club in alpha such that F restriction C is strictly increasing. We prove that it is consistent to have a cardinal theta such that strong non-reflection and weak reflection each hold on an unbounded set of cardinals less than theta .

Source: http://arxiv.org/abs/math/9504221v1

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Sep 20, 2013
09/13

by
Justin Tatch Moore; Michael Hrušák; Mirna Džamonja

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We will present a collection of guessing principles which have a similar relationship to $\diamond$ as cardinal invariants of the continuum have to $\CH$. The purpose is to provide a means for systematically analyzing $\diamond$ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of $\CH$ and $\diamond$ in models such as those of Laver, Miller, and Sacks.

Source: http://arxiv.org/abs/math/0608641v1

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Jun 30, 2018
06/18

by
James Cummings; Mirna Džamonja; Menachem Magidor; Charles Morgan; Saharon Shelah

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We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $\kappa$...

Topics: Mathematics, Logic

Source: http://arxiv.org/abs/1403.6795